The most promising ground-based technique for determining the mass of a minor planet is the observation of the perturbations it induces in the motion of another minor planet. This method requires careful observation of both minor planets over extended periods of time. The mass of (1) Ceres has been determined from the perturbations on (348) May, which made three close approaches to Ceres at intervals of 46 years between 1891 and 1984. The motion of May is clearly influenced by Ceres, and by using different test masses for Ceres, a search was made to determine the mass of Ceres that minimizes the residuals in the observations of May

Williams, Gareth V.

English

NASA Technical Reports Server

Asteroids, Comets, Meteors 1991, pp. 641-643
Lunar and Planetary Institute, Houston, 1992
w/eg-
THE MASS OF (1) CERES FROM
PERTURBATIONS ON (348) MAY
Gareth V. Williams
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, U.S.A.
641
ABSTRACT
The most promising ground-based technique for determining the mass of s minor planet is the
observation of the perturbations it induces in the motion of another minor planet. This method
requires cv.reful observation of both minor planets over extended periods of time. The mass of (1)
Ceres has been determined from the perturbations on (348) May, which made three close approaches
to Ceres at intervals of 46 years between 1891 and 1984. The motion of May is clearly influenced by
Ceres, and by Using different test masses for Ceres, a search has been made to determine the mass
of Ceres that minimizes the residuals in the observations of May.
Introduction
The masses of the largest minor planets are rather poorly known. Traditionally, the masses of the
major planets were obtained by observing their mutual perturbations or by observing their satellite systems.
These methods are not easily applicable to minor planets: no minor planets are known to have satellites;
the minor-planet perturbations on any of the major planets are negligible (from a ground-based viewpoint,
except by tracking spacecraft); and the mutual minor-planet perturbations generally are small. However, the
mutual-perturbation method, when applied to suitable objects, is the best ground-based method currently
available for determimng minor-planet masses. The best circumstances for using this method occur when one
of the minor planets is large compared to the other, when the two objects make close, periodic approaches
to each other, and when both have long observational histories.
A suitable pair of objects with long observational histories that make periodic close approaches to one
another are (1) Ceres and (348) May: to 0.0816 AU in April 1938 and to 0.0424 AU in September 1984 (a
moderately close-approach to 0.2006 AU occurred in December 1891, just prior to May's discovery). The
orbital inclinations and nodal longitudes of Ceres and May are broadly similar, and since both orbits are of
low eccentricity with semi-major axes differing by only 0.2 AU, encounters between the two tend to be quite
prolonged: during the 1938 encounter May was within 0.1 AU of Ceres from December 1937 until July 1938;
in 1984 the corresponding dates are February 1984 and March 1985. In addition, May is so small that its
perturbing effect on Ceres can be neglected.
The first direct measurement of the mass of a minor planet was made by Hertz (1966), who, noting
that (4) Vesta and (197) Arete approached each other to within a few hundredths of an AU every 18 years,
determined the mass of Vesta to be (1.17 4. 0.10) x 10-1°M®. This value was later revised to (1.20 4. 0.08) ×
10-1°M® (Hertz, 1968). Prior to this determination, the masses of minor planets were estimated on the
basis of measured or presumed diameters and presumed densities.
The suggestion that the mass of Ceres, Me, might be determined from the perturbations it induced in
the motion of a smaller body was made by Gauss. In 1802, shortly after the discovery of Pallas, he noted
the close approach of the orbits of Ceres and Pallas at the descending node of the latter, and expressed the
hope that the masses of both bodies might be determined from their gravitional influences on each other.
From consideration of the perturbations on Pallas by Ceres, Sehubart (1970) derived Mc = (6.7 4-0.4) x
10-1°M®. From the perturbations by Ceres on Vesta, Schubart (1971) derived Mc = 5.1 x 10-1°M®, and
combined this result and his 1970 value to obtain M_ = (6.0 4- 0.7) x 10-1°M®. Schubart (1974), using a
slightly different set of Pallas observations, refined this value to M, = (5.9 4- 0.3) × 10-t°M®. Landgraf
1988), again considering perturbations on Pailas, found Mc = (5.0 4. 0.3) × 10-1°M®. Standish and Hellings
1989), considering the perturbations on Mars determined from the highly accurate ranging data from the
Viking Inhalers, found Mc = (5.0 4. 0.2) x 10-1°M®.
For some time, E. Bowell, Lowell Observatory, has been recalculating the orbits of the old numbered
minor planets on a regular basis. Bowell was unsuccessful in his attempt to represent the motion of May by
considering perturbations by Mercury to Pluto, and he brought the matter to the attention of the author at
the Minor Planet Center (hereafter referred to as MPC). After several attempts to obtain a reasonable orbit
failed, the author realized that Ceres might be perturbing May significantly. Since the presumed effect of
https://ntrs.nasa.gov/search.jsp?R=19930010073 2020-03-17T08:17:24+00:00Z
642 Asteroids, Comets, Meteors 1991
Ceres was quite large (more than 100" in right ascension between 1892 and 1989), it was reasoned that the
perturbations on May might be a rather sensitive indicator of the mass of Ceres. This hypothesis could not
be tested until development versions of perturbation programs designed to work in the J2000.0 system were
completed in late 1990, as the 81950.0 versions in use previously at the MPC were not designed to handle
Ceres as a perturbing body.
The Observations
(348) May was discovered by A. Charlois in Nice on November 18, 1802. It was followed until January
1893 and was recovered by J. Palisa in Vienna in March 1894. It was followed there and at Nice for a month,
but was not observed again until 1905. Observations seem to have been made at almost every opposition
subsequent to 19!0 .... _ . ........ := : :_:: ....... . ......
It is known that the archive of minor-planet observations at the MPC is fragmentary for observations
made before 1939. A project to include more pre-1939 material in the archive is progressing. Efforts are
also being made to encourage the remeasurement of plates for which only approximate positions have been
previously reported.
A literature search was undertaken to obtain pre-1939 observations of May, so the observed arc could
be extended back to the discovery opposition. Good series of observations made at Algiers were located. It
was possible to rereduce the micrometric observations using the planet-minus-comparison offsets as given in
the original publication and modern comparison-star positions. The rereduction of micrometer measures is
a tedious task to do by hand, so the process was automated. The program that was written to accomplish
this task, COMPSTAR, will also be used to handle the automatic rereduction of micrometric measures that
are encountered i_the course of the long-term project to expand the MPC archives.
A number of positions of (348) May in the MPC archive were approximate p_sitsitions_-Such- pd_G0ns
are useless for orbit determination, and their only purpose is to show that there is a plate available for
remeasurement. Precise measurements were desired for the series of approximate positions ob_tained at the
Goethe Link Observatory, Indiana, and Turku, Finland. The plates taken in Indiana are now archived at the
Lowell Observatory. The Indiana plates containing images of May were remeasured by B. A. Skiff, following
a request to E. Bowell, and the relevant Turku plates were remeasured by A. Niemi, following a request to
L. Oterma. A search of the Harvard plate-archives by C. Y. Shao led to the identificativn of a number of
images of May from the 1920s and 1930s, and measurement of these images was undertaken by A. J. Noymer.
At the request of the author May was put on the observing program of the Oak Ridge Observatory. A
number of observations at the 1991 opposition were obtained, extending over an arc of three mo_ A new
orbit for May, computed in the :12000.0 system, published recently in the Minor Plane¢ Circulars (Williams,
1991) includes these new observations.
A number of different orbits were calculated for May, considering the perturbations by MercUry to
Neptune plus Ceres, using a different mass for Ceres in each orbit. The same observations were used in each
calculation: 75 observations at 29 oppositions, 1892-1991. These observations showed reasonable residuals
from a preliminary orbit calculation. = _= .............
The first mass used was the 1976 IAU-recommended value, 5.9 x 10-t°M@. Solutions were also done
using masses ranging from (in units of 10-1°M®) 4.0 to 5.6. The r.m.s, residuals for each orbit solution are
gxven below:
Mass (x lO-l°M®) 0 4.0 4.2
r.m.s, residual (") 2.9T 1.23 1.19
Mass (x IO-i0M®)" 5.2 5.4 5.6
r.m.s, residual (") 1.16 1.19 1.23
4.4 4.6 4.7 4.8
...... I
1.16 1.15 1.14 1.14
5.9
1.31
4.9 5.0
1.14 1.15
It can be seen from the table that the IAU-recommended value is too large. The mass of Ceres and its
mean error were determined by the method of Herget (1972): this gave (4.80 + 0.22) x 10-1°M_.
As a test to see whether the motion of May was affected by any large minor planet other than Ceres,
the influence of (511) Davida was examined (May and Davida can approach to within 0.05 AU). Assuming
that Ceres and Davida have the same density, and using the diameters given by Tedesco (1989), a reasonable
estimate for the mass of Davida was 3 x 10- xl M®. The orbit of May was recalculated twice: once considering
!
Asteroids, Comets, Meteors 1991 643
perturbations by Venus to Neptune plus Ceres; and once including perturbations by Davids. Comparison
of the residuals of both solutions showed small differences of-_ 0'f4 in right ascension by 1892. This is only
0.4% of the difference caused by Ceres and does not significantly alter the estimate of the error of the derived
mass of Ceres.
Preliminary Conclusions and Further Work
A preliminary determination of the mass of (1) Ceres, deduced from the perturbations on (348) May, is
(4.80 4- 0.22) x 10-t°M®.
A request has been made to L. D. Schmadel and S. R6ser of the Astronomisches Rechen-Institnt,
Heidelberg, to secure the remeasurement of the relevant plates taken by Wolf and Reinmuth.
A search has also been made to locate other mi-
nor planets that approach Ceres periodically. Many ob-
jects were found to have orbits that approach Ceres very
closely. This extensive list can be shortened by consid-
ering the observed arc of each object, whether there are
any approaches to Ceres during the observed arc and, if
there are, the number and duration of these approaches.
Although this is quite time-consuming, several prom-
ising candidates have been identified, and study of the
motion oftheseobjectsmay help to constrainthe mass
of Ceres further.
A recentindependent study of the motion of(203)
Pompeja (Goffin_1991) determined the mass of Ceres
to be (4.7+0.3) x 10-1°M®. Although ithas been clear
for some time that the influenceof Pluto isnegligible
on the motion of most minor planetsand that Ceres
should be included as a perturbing body when orbits
are computed with fullperturbations,itisimportant
that allorbitcomputations are done using a standard
Object Date of Ami n
approach (AU)
(32) Pomona
(58) Concordia
(84) Klio
(91) Aegina
(135) Hertha
(165) Loreley
(203) Pompeja
(255) Oppavia
(324) Bamberga
1975 Nov.
1929 Sept.
1990 Mar.
1973 Sept.
1907 Feb.
1909 May
1937 Jan.
1964 Sept.
1948 Aug.
1969 Feb.
1955 Nov.
1958 Apr.
1944 Mar.
0.0250
0.0408
0.0417
0.0332
0.0529
0.0824
0.0387
0.0966
0.0165
0.0781
0.0804
0.0418
0.0207
value for the mass of Ceres. Commission 20 of the IAU, at a sessionat the 1991 IAU General Assembly in
Buenos Aires,suggested the use of5.0 × 10-1°Me as the mass ofCeres. This isthe value that isnow inuse
at the MPC.
References
Gofl_n E. (1991) Astron. gJ Astrophys., 249, 563.
Herget P. (1972) On The Determination of Planetary Masses. In IAU Syrup. No. _5: The Motion, Evol_.tion
of Orbits, and Origin of Comets (G. A. Chebotarev, E. I. Kazimirchak-Polonskaya and B. G. Marsden,
eds.), pp. 244-245.
Hertz H. G. (1966) IAU Circular 1983.
Hertz H. G. (1968) Science, 160, 299.
LandgrafW. (1988) Astron. _ Astrophys., 191, 161.
Schubart J. (1970) IAU Circular 2268.
Schubart J. (1971) Asteroid Masses and Densities.In PltysicalStudies of Minor Plane_s (T. Gehrels,ed.),
pp. 33-39.
Schubart 3.(1974) Astron. _ Astropltys.,30, 289.
Standish E. M. and HellingsR. W. (1989) Icarus,80, 326.
Tedesco E. F. (1989) Asteroid Magnitudes, UBV Colors,and IRAS Albedos and Diameters. In AsteroidsII
(R. P. Binzel,T. Gehrels,M. S. Matthews, eds.),1090-1138.
Williams G. V. (1991) Minor Planet Circular19274.
Z? 2
m


The mass of (1) Ceres from perturbations on (348) May

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